Effective moduli of rocks predicted by the Kuster–Toksöz and Mori–Tanaka models

Natural rocks are polymineral composites with complex microstructures. Such strong heterogeneities significantly affect the estimation of effective moduli by some theoretical models. First, we have compared the effective moduli of isotropic rocks predicted by the Kuster–Toksöz (KT) model and the Mori–Tanaka (MT) model. The widely used KT model only has finite precision in many cases because of its assumption that is restricted to the first-order scattering approximation. However, the MT model based on the Eshelby tensor in mesomechanics has the advantage of predicting effective moduli of rocks, especially when the volume fraction of embedded inclusions is sufficiently large. In addition, the MT model can be used to predict the effective modulus of anisotropic rocks, but the KT model cannot. For a certain kind of shale or tight sandstones, which are viewed as isotropic composites, both the models work well. For the medium containing spherical pores, both the models produce the same results, whereas for ellipsoidal pores the MT model is more accurate than the KT model, validated by the finite element simulations. In what follows, the applicable ranges of simplified formulas for pores with needle, coin and disk shapes, widely used in engineering, are quantitatively given based on the comparison with the results according to the reduced ellipsoidal formulas of the MT and KT models. These findings provide a comprehensive understanding of the two models in calculating the effective modulus of rocks, which are beneficial to such areas as petroleum exploration and exploitation, civil engineering, and geophysics.